CHAPTER 1

Functions EXERCISE SET 1.1 1. (a) around 1943 (b) 1960; 4200 (c) no; you need the year’s population (d) war; marketing techniques (e) news of health risk; social pressure, antismoking campaigns, increased taxation 2. (a) 1989; \$35,600 (b) 1975, 1983; \$32,000 (c) the ﬁrst two years; the curve is steeper (downhill) 3. (a) −2.9, −2.0, 2.35, 2.9 (d) −1.75 ≤ x ≤ 2.15

(b) none (c) y = 0 (e) ymax = 2.8 at x = −2.6; ymin = −2.2 at x = 1.2

4. (a) x = −1, 4 (d) x = 0, 3, 5

(b) none (c) y = −1 (e) ymax = 9 at x = 6; ymin = −2 at x = 0

5. (a) x = 2, 4

(b) none

(c) x ≤ 2; 4 ≤ x

(d) ymin = −1; no maximum value

6. (a) x = 9

(b) none

(c) x ≥ 25

(d) ymin = 1; no maximum value

7. (a) Breaks could be caused by war, pestilence, ﬂood, earthquakes, for example. (b) C decreases for eight hours, takes a jump upwards, and then repeats. 8. (a) Yes, if the thermometer is not near a window or door or other source of sudden temperature change. (b) No; the number is always an integer, so the changes are in movements (jumps) of at least one unit. 9. (a) The side adjacent to the building has length x, so L = x + 2y. Since A = xy = 1000, L = x + 2000/x. (b) x > 0 and x must be smaller than the width of the building, which was not given. (c)

(d) Lmin ≈ 89.44 ft

120

20

80 80

10. (a) V = lwh = (6 − 2x)(6 − 2x)x (c)

(b) From the ﬁgure it is clear that 0 < x < 3. (d) Vmax ≈ 16 in3

20

0

3 0

1

2

Chapter 1

11. (a) V = 500 = πr2 h so h =

500 . Then πr2

C = (0.02)(2)πr2 + (0.01)2πrh = 0.04πr2 + 0.02πr = 0.04πr2 +

7

500 πr2

10 ; Cmin ≈ 4.39 at r ≈ 3.4, h ≈ 13.8. r 1.5

6 4

10 (b) C = (0.02)(2)(2r)2 + (0.01)2πrh = 0.16r2 + . Since r 0.04π < 0.16, the top and bottom now get more weight. Since they cost more, we diminish their sizes in the solution, and the cans become taller.

7

1.5

5.5 4

(c) r ≈ 3.1 cm, h ≈ 16.0 cm, C ≈ 4.76 cents 12. (a) The length of a track with straightaways of length L and semicircles of radius r is P = (2)L + (2)(πr) ft. Let L = 360 and r = 80 to get P = 720 + 160π = 1222.65 ft. Since this is less than 1320 ft (a quarter-mile), a solution is possible. (b) P = 2L + 2πr = 1320 and 2r = 2x + 160, so L = 12 (1320 − 2πr) = 12 (1320 − 2π(80 + x)) = 660 − 80π − πx.

450

0

100 0

(c) The shortest straightaway is L = 360, so x = 15.49 ft. (d) The longest straightaway occurs when x = 0, so L = 660 − 80π = 408.67 ft.

EXERCISE SET 1.2 1. (a) f (0) = 3(0)2 −2 = −2; f (2) = 3(2)2 −2 = 10; f (−2) = 3(−2)2 −2 = 10; f (3) = 3(3)2 −2 = 25; √ √ f ( 2) = 3( 2)2 − 2 = 4; f (3t) = 3(3t)2 − 2 = 27t2 − 2 √ √ (b) f (0) = 2(0) = 0; f (2) = 2(2) = 4; f (−2) = 2(−2) = −4; f (3) = 2(3) = 6; f ( 2) = 2 2; f (3t) = 1/3t for t > 1 and f (3t) = 6t for t ≤ 1. −1 + 1 π+1 −1.1 + 1 −0.1 1 3+1 = 2; g(−1) = = 0; g(π) = ; g(−1.1) = = = ; 3−1 −1 − 1 π−1 −1.1 − 1 −2.1 21 2 2 t −1+1 t g(t2 − 1) = 2 = 2 t −1−1 t −2 √ √ (b) g(3) = 3 + 1 = 2; g(−1) = 3; g(π) = π + 1; g(−1.1) = 3; g(t2 − 1) = 3 if t2 < 2 and √ g(t2 − 1) = t2 − 1 + 1 = |t| if t2 ≥ 2.

2. (a) g(3) =

√ √ 3. (a) x = 3 (b) x ≤ − 3 or x ≥ 3 (c) x2 − 2x + 5 = 0 has no real solutions so x2 − 2x + 5 is always positive or always negative. If x = 0, then x2 − 2x + 5 = 5 > 0; domain: (−∞, +∞). (d) x = 0 (e) sin x = 1, so x = (2n + 12 )π, n = 0, ±1, ±2, . . .

Exercise Set 1.2

3

4. (a) x = − 75 (b) x − 3x2 must be nonnegative; y = x − 3x2 is a parabola that crosses the x-axis at x = 0, 13 and opens downward, thus 0 ≤ x ≤ 13 x2 − 4 > 0, so x2 − 4 > 0 and x − 4 > 0, thus x > 4; or x2 − 4 < 0 and x − 4 < 0, thus (c) x−4 −2 < x < 2 (d) x = −1 (e) cos x ≤ 1 < 2, 2 − cos x > 0, all x 5. (a) x ≤ 3

(b) −2 ≤ x ≤ 2

(c) x ≥ 0

(d) all x

(e) all x

6. (a) x ≥

(b) − 32 ≤ x ≤

(c) x ≥ 0

(d) x = 0

(e) x ≥ 0

2 3

3 2

7. (a) yes (c) no (vertical line test fails)

(b) yes (d) no (vertical line test fails)

8. The sine of θ/2 is (L/2)/10 (side opposite over hypotenuse), so that L = 20 sin(θ/2). 9. The cosine of θ is (L − h)/L (side adjacent over hypotenuse), so h = L(1 − cos θ). 10.

11.

T

t

12.

h

w

t

t 5

10

15

13. (a) If x < 0, then |x| = −x so f (x) = −x + 3x + 1 = 2x + 1. If x ≥ 0, then |x| = x so f (x) = x + 3x + 1 = 4x + 1;  2x + 1, x < 0 f (x) = 4x + 1, x ≥ 0 (b) If x < 0, then |x| = −x and |x − 1| = 1 − x so g(x) = −x + 1 − x = 1 − 2x. If 0 ≤ x < 1, then |x| = x and |x − 1| = 1 − x so g(x) = x + 1 − x = 1. If x ≥ 1, then |x| = x and |x − 1| = x − 1 so g(x) = x + x − 1 = 2x − 1;  x<0  1 − 2x, 1, 0≤x<1 g(x) =  2x − 1, x≥1 14. (a) If x < 5/2, then |2x − 5| = 5 − 2x so f (x) = 3 + (5 − 2x) = 8 − 2x. If x ≥ 5/2, then |2x − 5| = 2x − 5 so f (x) = 3 + (2x − 5) = 2x − 2;  8 − 2x, x < 5/2 f (x) = 2x − 2, x ≥ 5/2 (b) If x < −1, then |x − 2| = 2 − x and |x + 1| = −x − 1 so g(x) = 3(2 − x) − (−x − 1) = 7 − 2x. If −1 ≤ x < 2, then |x − 2| = 2 − x and |x + 1| = x + 1 so g(x) = 3(2 − x) − (x + 1) = 5 − 4x. If x ≥ 2, then |x − 2| = x − 2 and |x + 1| = x + 1 so g(x) = 3(x − 2) − (x + 1) = 2x − 7;  x < −1  7 − 2x, 5 − 4x, −1 ≤ x < 2 g(x) =  2x − 7, x≥2

4

Chapter 1

15. (a) V = (8 − 2x)(15 − 2x)x (b) −∞ < x < +∞, −∞ < V < +∞ (c) 0 < x < 4 (d) minimum value at x = 0 or at x = 4; maximum value somewhere in between (can be approximated by zooming with graphing calculator) (b) θ = nπ + π/2 for n an integer, −∞ < n < ∞ (d) 3000 ft

16. (a) x = 3000 tan θ (c) 0 ≤ θ < π/2, 0 ≤ x < +∞

6000

6

0 0

17. (i)

x = 1, −2 causes division by zero

(ii)

g(x) = x + 1, all x

18. (i)

x = 0 causes division by zero

(ii)

g(x) =

19. (a) 25◦ F

(b) 2◦ F

x + 1 for x ≥ 0 (c) −15◦ F

20. If v = 48 then −60 = WCI = 1.6T − 55; thus T = (−60 + 55)/1.6 ≈ −3◦ F. √ 21. If v = 8 then −10 = WCI = 91.4 + (91.4 −√T )(0.0203(8) − 0.304 8 − 0.474); thus T = 91.4 + (10 + 91.4)/(0.0203(8) − 0.304 8 − 0.474) and T = 5◦ F 22. The WCI is given by three formulae, but the ﬁrst√and third don’t work√with the data. Hence −15 = WCI = 91.4 + (91.4 − 20)(0.0203v − 0.304 v − 0.474); set x = v so that v = x2 and obtain 0.0203x2 − 0.304x − 0.474 + (15 + 91.4)/(91.4 − 20) = 0. Use the quadratic formula to ﬁnd the two roots. Square them to get v and discard the spurious solution, leaving v ≈ 25. 23. Let t denote time in minutes after 9:23 AM. Then D(t) = 1000 − 20t ft.

EXERCISE SET 1.3 1. (e) seems best, though only (a) is bad. y

2. (e) seems best, though only (a) is bad and (b) is not good.

0.5

3. (b) and (c) are good; (a) is very bad. y

y 0.5

14

x -1

13

1 12

x -0.5

-1

1

-0.5

-1

x 1

Exercise Set 1.3

5

5. [−3, 3] × [0, 5]

4. (b) and (c) are good; (a) is very bad.

6. [−4, 2] × [0, 3]

y

y

y -11 -12 x

-13

–2

x 2

-14 x -2

-1

0

1

2

7. (a) window too narrow, too short (c) good window, good spacing

(b) window wide enough, but too short (d) window too narrow, too short

y x -5

5

10

20

-200

-400

(e) window too narrow, too short 8. (a) window too narrow (c) good window, good tick spacing

(b) window too short (d) window too narrow, too short

y 50 -16

-8 -4

x 4

-100

-250

(e) shows one local minimum only, window too narrow, too short 9. [−5, 14] × [−60, 40]

10. [6, 12] × [−100, 100]

y

11. [−0.1, 0.1] × [−3, 3] y

y

40

100

3 2

20 x -5

5

x

10

10

x

12

-0.1

0.1

-50 -2 -60

12. [−1000, 1000] × [−13, 13]

13. [−250, 1050] × [−1500000, 600000]

y

y -1000

10 5 x -1000

1000 -5

-500000

x 1000

6

Chapter 1

14. [−3, 20] × [−3500, 3000]

15. [−2, 2] × [−20, 20]

16. [1.6, 2] × [0, 2]

y

y

y 2

20 1000 10

x 5

10

x

15

-2

-1000

1

-1

1

2 0.5

-10

x

-2000

-20

1.6 1.7

1.8

2

18. depends on graphing utility

17. depends on graphing utility y

y

6

6

4

4

2

2

x

-4

2

2

4

-2

-2

19. (a) f (x) =

x

-4 -2

4

√ (b) f (x) = − 16 − x2

16 − x2 y

y -4

3

2 -1

1

-3

x -2

2

4

y

(c)

x

-2

(d)

4

y

2 x -2

2 1

-2

x 1

4

(e) No; the vertical line test fails. 

20. (a) y = ±3

√ (b) y = ± x2 + 1

1 − x2 /4 y

y 4

3

2 x x -2

2

-4

-2

2 -2 -4

-3

4

4

Exercise Set 1.3

7

y

21. (a)

y

(b) 1

1

x

x -1

1

1 y

(c)

(d)

2

y 1

x -1

1

-1

(e)

1

x

c y

y

(f )

1

x

c

C

2c

x -1

1

–1 -1 y

22.

1

x -1

1

23. The portions of the graph of y = f (x) which lie below the x-axis are reﬂected over the x-axis to give the graph of y = |f (x)|. 24. Erase the portion of the graph of y = f (x) which lies in the left-half plane and replace it with the reﬂection over the y-axis of the portion in the right-half plane (symmetry over the y-axis) and you obtain the graph of y = f (|x|). 25. (a) for example, let a = 1.1

y

(b) 3

y 2 1 x x

a 1

26. They are identical.

2

3

y

27. 15

y

10 5 x

x -1

1

2

3

8

Chapter 1

28. This graph is very complex. We show three views, small (near the origin), medium and large: y

(a)

y

(b)

y

(c)

x 10

x -50

2

-1

1000

-1

x 40

y

29. (a)

y

(b) 1 0.5

1

x -3

-1 -0.5

1

3

x -1

-3

1

3

y

(c)

(d)

y

1.5 1

0.5

x

-1

1

2

0.5

3

x -2

-1

1

y

30.

1 x 2

31. (a) stretches or shrinks the graph in the y-direction; reﬂects it over the x-axis if c changes sign y

(b) As c increases, the parabola moves down and to the left. If c increases, up and right.

c=2

4

8

y

(c) The graph rises or falls in the y-direction with changes in c. y

c=2

6

c=1 c=1

2

4

x -1

2

1

2

-2 c = –1.5

c=2 c = 0.5 c = -1

8

-1

x c = –1.5 x 2

-2

1

2

Exercise Set 1.4

9

y

32. (a) 2

x 1

2

–2

(b) x-intercepts at x = 0, a, b. Assume a < b and let a approach b. The two branches of the curve come together. If a moves past b then a and b switch roles. y

y

y

3

3

3

1

1 1

-1

2

3

-1

a = 0.5 b = 1.5

-3

-3

33. The curve oscillates between the lines y = x and y = −x with increasing rapidity as |x| increases.

1

x

x 1

2

3

x 2

-1

a=1 b = 1.5

a = 1.5 b = 1.6

-3

34. The curve oscillates between the lines y = +1 and y = −1, inﬁnitely many times in any neighborhood of x = 0.

y

y

30 x

10 -30

-10 -10

x 10

–2

4

30 –1

-30

EXERCISE SET 1.4 y

1. (a)

(b)

1

x -1

1

0

y 2

1

2 x 1

-1 y

(c)

2 y

(d)

1

3

2 x -1

1

2

x -4

-2

2

3

10

Chapter 1

2. (a)

y -3

y

(b)

x 1

1 x -1 y

(c)

1 y

(d)

1

1 x 1

-1

x 1

-1 y

3. (a)

y

(b)

1

1 x

-2

-1

1

x

2 1

-1 -1 1 y

(c)

y

(d)

1

x -1

1

2

x

3

-1

1

3

-1

-1

4.

2

y

5. Translate right 2 units, and up one unit.

1 x

y

2 10

x -2

6. Translate left 1 unit, reﬂect over x-axis, and translate up 2 units. y

2

6

7. Translate left 1 unit, stretch vertically by a factor of 2, reﬂect over x-axis, translate down 3 units. y -6

x

-2 -20

1 -60 –2

x 2

6

Exercise Set 1.4

11

8. Translate right 3 units, compress vertically by a factor of 12 , and translate up 2 units.

9. y = (x + 3)2 − 9; translate left 3 units and down 9 units.

10. y = (x + 3)2 − 19; translate left 3 units and down 19 units.

y

y

15

y

x

–4

10

–5 x -8

-4 -2

2 -5

2 x 4

11. y = −(x − 1)2 + 2; translate right 1 unit, reﬂect over x-axis, translate up 2 units. y

12. y = 12 [(x − 1)2 + 2]; translate left 1 unit and up 2 units, compress vertically by a factor of 12 .

13. Translate left 1 unit, reﬂect over x-axis, translate up 3 units. y 2

y

2 x -2 -1 -2

1

1

3 2

x 2

1 -6

8

x 1

14. Translate right 4 units and up 1 unit. y

15. Compress vertically by a factor of 12 , translate up 1 unit.

16. Stretch vertically by √ a factor of 3 and reﬂect over x-axis.

y 4

y

x

2

2 -1

1 x 4

10

x 1

17. Translate right 3 units. y

2

3

18. Translate right 1 unit and reﬂect over x-axis.

10

y

19. Translate left 1 unit, reﬂect over x-axis, translate up 2 units. y

x 4

2

6 -2

-10

x

6

2 -4

2 -3

-1-2 -6

x 1

2

12

Chapter 1

20. y = 1 − 1/x; reﬂect over x-axis, translate up 1 unit.

21. Translate left 2 units and down 2 units. y

y x

5 -4

-2

x 2

-2

-5

22. Translate right 3 units, reﬂect over x-axis, translate up 1 unit.

23. Stretch vertically by a factor of 2, translate right 1 unit and up 1 unit.

y

y 4

1 x 5 -1

2

x 2

24. y = |x − 2|; translate right 2 units. y

25. Stretch vertically by a factor of 2, reﬂect over x-axis, translate up 1 unit.

2

y 3

1 x 2

1 x

4 -2

2 -1

26. Translate right 2 units and down 3 units. y

27. Translate left 1 unit and up 2 units.

28. Translate right 2 units, reﬂect over x-axis.

y

x

y

3

2

1 x

–2

4 1 x -3

-1

–1

1

 y

29. (a)

(b) y =

2

x -1

1

0 if x ≤ 0 2x if 0 < x

Exercise Set 1.4

13

y

30.

x 2 –5

31. (f + g)(x) = x2 + 2x + 1, all x; (f − g)(x) = 2x − x2 − 1, all x; (f g)(x) = 2x3 + 2x, all x; (f /g)(x) = 2x/(x2 + 1), all x 32. (f + g)(x) = 3x − 2 + |x|, all x; (f − g)(x) = 3x − 2 − |x|, all x; (f g)(x) = 3x|x| − 2|x|, all x; (f /g)(x) = (3x − 2)/|x|, all x = 0 √ √ 33. (f + g)(x) = 3 x − 1, x ≥ 1; (f − g)(x) = x − 1, x ≥ 1; (f g)(x) = 2x − 2, x ≥ 1; (f /g)(x) = 2, x > 1 34. (f + g)(x) = (2x2 + 1)/[x(x2 + 1)], all x = 0; (f − g)(x) = −1/[x(x2 + 1)], all x = 0; (f g)(x) = 1/(x2 + 1), all x = 0; (f /g)(x) = x2 /(x2 + 1), all x = 0 35. (a) 3

(b) 9

(c) 2

(d) 2

36. (a) π − 1

(b) 0

(c) −π 2 + 3π − 1

(d) 1

37. (a) t4 + 1

(b) t2 + 4t + 5

(c) x2 + 4x + 5

(f ) x2 + 1

(g) x + 1

(e) x2 + 2xh + h2 + 1 √

5s + 2 √ (e) 4 x

38. (a)

(b)

√

√ (c) 3 5x √ (g) 1/ 4 x

x+2

(f ) 0

1 +1 x2 (h) 9x2 + 1 (d)

√ (d) 1/ x (h) |x − 1|

39. (f ◦ g)(x) = 2x2 − 2x + 1, all x; (g ◦ f )(x) = 4x2 + 2x, all x 40. (f ◦ g)(x) = 2 − x6 , all x; (g ◦ f )(x) = −x6 + 6x4 − 12x2 + 8, all x 41. (f ◦ g)(x) = 1 − x, x ≤ 1; (g ◦ f )(x) = 42. (f ◦ g)(x) =

√

x2 + 3 − 3, |x| ≥

1 − x2 , |x| ≤ 1

6; (g ◦ f )(x) =

x, x ≥ 3

43. (f ◦ g)(x) =

1 1 1 1 , x = , 1; (g ◦ f )(x) = − − , x = 0, 1 1 − 2x 2 2x 2

44. (f ◦ g)(x) =

x 1 , x = 0; (g ◦ f )(x) = + x, x = 0 x2 + 1 x

45. x−6 + 1 47. (a) g(x) =

46. √

x, h(x) = x + 2

x x+1 (b) g(x) = |x|, h(x) = x2 − 3x + 5

48. (a) g(x) = x + 1, h(x) = x2

(b) g(x) = 1/x, h(x) = x − 3

49. (a) g(x) = x2 , h(x) = sin x

(b) g(x) = 3/x, h(x) = 5 + cos x

14

Chapter 1

50. (a) g(x) = 3 sin x, h(x) = x2

(b) g(x) = 3x2 + 4x, h(x) = sin x

51. (a) f (x) = x3 , g(x) = 1 + sin x, h(x) = x2

(b) f (x) =

52. (a) f (x) = 1/x, g(x) = 1 − x, h(x) = x2

(b) f (x) = |x|, g(x) = 5 + x, h(x) = 2x

53.

2 1

x, g(x) = 1 − x, h(x) =

√ 3

54. {−2, −1, 0, 1, 2, 3}

y x

-3 -2 -1 -1

1

2

3

-2 -3 -4

55. Note that f (g(−x)) = f (−g(x)) = f (g(x)), so f (g(x)) is even.

56. Note that g(f (−x)) = g(f (x)), so g(f (x)) is even. y 3 g( f (x))

y f (g(x))

1 1 –3

–1 –1

x

–3

1

–1 –1

x 1

3

–2

–3

57. f (g(x)) = 0 when g(x) = ±2, so x = ±1.4; g(f (x)) = 0 when f (x) = 0, so x = ±2. 58. f (g(x)) = 0 at x = −1 and g(f (x)) = 0 at x = −1 59.

6xh + 3h2 3(x + h)2 − 5 − (3x2 − 5) = 6x + 3h; = h h 3w2 − 5 − (3x2 − 5) 3(w − x)(w + x) = 3w + 3x = w−x w−x

60.

(x + h)2 + 6(x + h) − (x2 + 6x) 2xh + h2 + 6h = = 2x + h + 6; h h w2 + 6w − (x2 + 6x) =w+x+6 w−x

61.

x − (x + h) −1 1/w − 1/x x−w 1 1/(x + h) − 1/x = = ; = =− h xh(x + h) x(x + h) w−x wx(w − x) xw

62.

x2 − (x + h)2 x2 − w2 2x + h 1/w2 − 1/x2 1/(x + h)2 − 1/x2 x+w = 2 = = − ; =− 2 2 2 2 2 2 2 h x h(x + h) x (x + h) w−x x w (w − x) x w

63. (a) the origin

(b) the x-axis

(c) the y-axis

(d) none

x

Exercise Set 1.4

15

y

64. (a)

(b)

y x

x

y

(c)

x

65. (a)

x f (x)

−3 1

−2 −5

−1 −1

0 0

1 −1

(b)

x f (x)

−3 1

−2 5

−1 −1

0 0

1 1

2 −5 2 −5

3 1 3 −1

y

66. (a)

(b)

y

x

67. (a) even

(b) odd

x

(c) odd

(d) neither

68. neither; odd; even 69. (a) f (−x) = (−x)2 = x2 = f (x), even (c) f (−x) = | − x| = |x| = f (x), even (e) f (−x) = (f )

(b) f (−x) = (−x)3 = −x3 = −f (x), odd (d) f (−x) = −x + 1, neither

x3 + x (−x)3 − (−x) = − = −f (x), odd 1 + (−x)2 1 + x2

f (−x) = 2 = f (x), even

70. (a) x-axis, because x = 5(−y)2 + 9 gives x = 5y 2 + 9 (b) x-axis, y-axis, and origin, because x2 − 2(−y)2 = 3, (−x)2 − 2y 2 = 3, and (−x)2 − 2(−y)2 = 3 all give x2 − 2y 2 = 3 (c) origin, because (−x)(−y) = 5 gives xy = 5 71. (a) y-axis, because (−x)4 = 2y 3 + y gives x4 = 2y 3 + y (b) origin, because (−y) =

x (−x) gives y = 3 + (−x)2 3 + x2

(c) x-axis, y-axis, and origin because (−y)2 = |x| − 5, y 2 = | − x| − 5, and (−y)2 = | − x| − 5 all give y 2 = |x| − 5

16

Chapter 1

72.

73.

3

-4

4

2

-3

3

–3

-2

74. (a) Whether we replace x with −x, y with −y, or both, we obtain the same equation, so by Theorem 1.4.3 the graph is symmetric about the x-axis, the y-axis and the origin. (b) y = (1 − x2/3 )3/2 (c) For quadrant II, the same; for III and IV use y = −(1 − x2/3 )3/2 . (For graphing it may be helpful to use the tricks that precede Exercise 29 in Section 1.3.) y

75.

y

76.

5

2

2 x

x 1

2

2 y

77. (a)

O

c

C

o

y

78. (a)

y

(b)

1

2

x O

c

C

o

x

y

(b)

1 1

x -1

1

x 2‚ 1 ‚ 3

-‚ 2 -1

y

(c)

(d)

y 1 x

3 C

c/2

x -1

1

79. Yes, e.g. f (x) = xk and g(x) = xn where k and n are integers. 80. If x ≥ 0 then |x| = x and f (x) = g(x). If x < 0 then f (x) = |x|p/q if p is even and f (x) = −|x|p/q if p is odd; in both cases f (x) agrees with g(x).

Exercise Set 1.5

17

EXERCISE SET 1.5 3−0 3 3 − (8/3) 1 0 − (8/3) 2 =− , =− , = 0−2 2 0−6 18 2−6 3 (b) Yes; the ﬁrst and third slopes above are negative reciprocals of each other.

1. (a)

2. (a)

−1 − (−1) −1 − 3 3−3 −1 − 3 = 0, = 2, = 0, =2 −3 − 5 5−7 7 − (−1) −3 − (−1)

(b) Yes; there are two pairs of equal slopes, so two pairs of parallel lines. 3. III < II < IV < I

4. III < IV < I < II

1 − (−5) −5 − (−1) 1 − (−1) = 2. Since the slopes connecting all pairs of points = 2, = 2, 1 − (−2) −2 − 0 1−0 are equal, they lie on a line. 2−5 4−5 1 4−2 = −1, = 3, = . Since the slopes connecting the pairs of points are (b) −2 − 0 0−1 −2 − 1 3 not equal, the points do not lie on a line.

5. (a)

6. The slope, m = −2, is obtained from (a) If x = 9 then y = 1. 7. The slope, m = 3, is equal to (a) If x = 5 then y = 14.

y−5 , and thus y − 5 = −2(x − 7). x−7 (b) If y = 12 then x = 7/2.

y−2 , and thus y − 2 = 3(x − 1). x−1 (b) If y = −2 then x = −1/3.

y−0 1 y−5 = or y = x/2. Also = 2 or y = 2x − 9. Solve x−0 2 x−7 simultaneously to obtain x = 6, y = 3.

8. (a) Compute the slopes:

5−0 2−0 and the second is . Since they are negatives of each other we 1−x 4−x get 2(4 − x) = −5(1 − x) or 7x = 13, x = 13/7.

9. (a) The ﬁrst slope is

10. (a) 27◦

(b) 135◦

(c) 63◦

(d) 91◦

11. (a) 153◦

(b) 45◦

(c) 117◦

(d) 89◦

√ 12. (a) m = tan φ = − 3/3, so φ = 150◦ 13. (a) m = tan φ =

3, so φ = 60◦

(b) m = tan φ = 4, so φ = 76◦ (b) m = tan φ = −2, so φ = 117◦

14. y = 0 and x = 0 respectively 15. y = −2x + 4

16. y = 5x − 3 6

2 –1

-1

2

1 0

–8

18

Chapter 1

17. Parallel means the lines have equal slopes, so y = 4x + 7. 12

18. The slope of both lines is −3/2, so y − 2 = (−3/2)(x − (−1)), or y = − 32 x + 12 4

-2 -1

1

1 0 –4

19. The negative reciprocal of 5 is −1/5, so y = − 15 x + 6. 12

20. The slope of x − 4y = 7 is 1/4 whose negative reciprocal is −4, so y − (−4) = −4(x − 3) or y = −4x + 8. 9

-9

9

0

0

18

–3

6−1 = −5, so −3 − (−2) y − 6 = −5(x − (−3)), or y = −5x − 9

4 − (4 − 7) = 11, 2−1 so y − (−7) = 11(x − 1), or y = 11x − 18

22. m =

21. m =

15

7

0

4 –3

0 –5

-9

23. (a) m1 = m2 = 4, parallel (b) m1 = 2 = −1/m2 , perpendicular (c) m1 = m2 = 5/3, parallel (d) If A = 0 and B = 0 then m1 = −A/B = −1/m2 , perpendicular; if A = 0 or B = 0 (not both) then one line is horizontal, the other vertical, so perpendicular. (e) neither 24. (a) m1 = m2 = −5, parallel (c) m1 = −4/5 = −1/m2 , perpendicular (e) neither

(b) m1 = 2 = −1/m2 , perpendicular (d) If B = 0, m1 = m2 = −A/B; if B = 0 both are vertical, so parallel

25. (a) m = (0 − (−3))/(2 − 0)) = 3/2 so y = 3x/2 − 3 (b) m = (−3 − 0)/(4 − 0) = −3/4 so y = −3x/4 26. (a) m = (0 − 2)/(2 − 0)) = −1 so y = −x + 2 (b) m = (2 − 0)/(3 − 0) = 2/3 so y = 2x/3

Exercise Set 1.5

19

27. (a) The velocity is the slope, which is

5 − (−4) = 9/10 ft/s. 10 − 0

(b) x = −4 (c) The line has slope 9/10 and passes through (0, −4), so has equation x = 9t/10 − 4; at t = 2, x = −2.2. (d) t = 80/9 28. (a) v =

5−1 = 2 m/s 4−2

(b) x − 1 = 2(t − 2) or x = 2t − 3

(c) x = −3

3 − (−1) 4 = − ft/s2 . 1−4 3 13 (c) v = 3 ft/s

29. (a) The acceleration is the slope of the velocity, so a = (b) v − 3 = − 43 (t − 1), or v = − 43 t +

13 3

30. (a) The acceleration is the slope of the velocity, so a = (b) v = 5 ft/s

(c) v = 4 ft/s

5 1 0−5 =− = − ft/s2 . 10 − 0 10 2 (d) t = 4 s

31. (a) It moves (to the left) 6 units with velocity v = −3 cm/s, then remains motionless for 5 s, then moves 3 units to the left with velocity v = −1 cm/s. 9 0−9 (b) vave = =− cm/s 10 − 0 10 (c) Since the motion is in one direction only, the speed is the negative of the velocity, so 9 save = 10 cm/s. 32. It moves right with constant velocity v = 5 km/h; then accelerates; then moves with constant, though increased, velocity again; then slows down. 33. (a) If x1 denotes the ﬁnal position and x0 the initial position, then v = (x1 − x0 )/(t1 − t0 ) = 0 mi/h, since x1 = x0 . (b) If the distance traveled in one direction is d, then the outward journey took t = d/40 h. Thus 2d 80t total dist = = = 48 mi/h. save = total time t + (2/3)t t + (2/3)t (c) t + (2/3)t = 5, so t = 3 and 2d = 80t = 240 mi round trip 34. (a) down, since v < 0

(b) v = 0 at t = 2

  if 0 ≤ t ≤ 10  10t 100 if 10 ≤ t ≤ 100 (b) v =   600 − 5t if 100 ≤ t ≤ 120

v

35. (a) 60

20 t 20

36.

(c) It’s constant at 32 ft/s2 .

80

x

t

20

Chapter 1

37. (a) y = 20 − 15 = 5 when x = 45, so 5 = 45k, k = 1/9, y = x/9 y

(b)

(c) l = 15 + y = 15 + 100(1/9) = 26.11 in. (d) If ymax = 15 then solve 15 = kx = x/9 for x = 135 lb.

0.6

0.2 x 2

6

38. (a) Since y = 0.2 = (1)k, k = 1/5 and y = x/5 y

(b)

(c) y = 3k = 3/5 so 0.6 ft. (d) ymax = (1/2)3 = 1.5 so solve 1.5 = x/5 for x = 7.5 tons

1

x 2

6

39. Each increment of 1 in the value of x yields the increment of 1.2 for y, so the relationship is linear. If y = mx + b then m = 1.2; from x = 0, y = 2, follows b = 2, so y = 1.2x + 2 40. Each increment of 1 in the value of x yields the increment of −2.1 for y, so the relationship is linear. If y = mx + b then m = −2.1; from x = 0, y = 10.5 follows b = 10.5, so y = −2.1x + 10.5 41. (a) With TF as independent variable, we have

TC − 100 5 0 − 100 = , so TC = (TF − 32). TF − 212 32 − 212 9

(b) 5/9 (c) Set TF = TC = 59 (TF − 32) and solve for TF : TF = TC = −40◦ (F or C). (d) 37◦ C 42. (a) One degree Celsius is one degree Kelvin, so the slope is the ratio 1/1 = 1. Thus TC = TK − 273.15. (b) TC = 0 − 273.15 = −273.15◦ C 43. (a)

p−1 5.9 − 1 = , or p = 0.098h + 1 h−0 50 − 0

44. (a)

133.9 − 123.4 R − 123.4 = , so R = 0.42T + 115. T − 20 45 − 20

(b) T = 32.38◦ C

45. (a)

r − 0.80 0.75 − 0.80 = , so r = −0.0125t + 0.8 t−0 4−0

(b) 64 days

(b) when p = 2, or h = 1/0.098 ≈ 10.20 m

46. (a) Let the position at rest be y0 . Then y0 + y = y0 + kx; with x = 11 we get y0 + kx = y0 + 11k = 40, and with x = 24 we get y0 + kx = y0 + 24k = 60. Solve to get k = 20/13 and y0 = 300/13. (b) 300/13 + (20/13)W = 30, so W = (390 − 300)/20 = 9/2 g.

Exercise Set 1.6

21

47. (a) For x trips we have C1 = 2x and C2 = 25 + x/4 C

(b) 2x = 25 + x/4, or x = 100/7, so the commuter pass becomes worthwhile at x = 15.

60

20 x 10

20

48. If the student drives x miles, then the total costs would be CA = 4000 + (1.25/20)x and CB = 5500 + (1.25/30)x. Set 4000 + 5x/80 = 5500 + 5x/120 and solve for x = 72, 000 mi.

EXERCISE SET 1.6 1. (a) y = 3x + b

(b) y = 3x + 6

y

(c)

y = 3x + 6

10

y = 3x + 2 y = 3x - 4 x

-2

2

-10

2. Since the slopes are negative reciprocals, y = − 13 x + b. 3. (a) y = mx + 2 (b) m = tan φ = tan 135◦ = −1, so y = −x + 2

y

(c)

m = 1.5 m=1

5 4 3

m = –1

1

x

-2

1

2

4. (a) y = mx (c) y = −2 + m(x − 1)

(b) y = m(x − 1) (d) 2x + 4y = C

5. (a) The slope is −1.

(b) The y-intercept is y = −1.

y

4

5

y

2 x 2

-1

1

x -1

1 -2 -3

2

-4

22

Chapter 1

(c) They pass through the point (−4, 2).

(d) The x-intercept is x = 1.

y

y 3

6

1 x

2 1

x -4

-6

2

-1

-2 -2

-3

(b) The y-intercept is y = −1/2.

6. (a) horizontal lines y

y 2 x x 2 -1

(c) The x-intercept is x = −1/2.

(d) They pass through (−1, 1). y

y (-1, 1) 1

1 x 1

x -2

1

7. Let the line be tangent to the circle at the point (x0 , y0 ) where x20 + y02 = 9. The slope of the tangent line is the negative reciprocal of y0 /x0 (why?), so m = −x0 /y0 and y = −(x0 /y0 )x + b.  9 − x0 x Substituting the point (x0 , y0 ) as well as y0 = ± 9 − x20 we get y = ±  . 9 − x20 8. Solve the simultaneous equations to get the point (−2, 1/3) of intersection. Then y =

1 3

+ m(x + 2).

9. The x-intercept is x = 10 so that with depreciation at 10% per year the ﬁnal value is always zero, and hence y = m(x − 10). The y-intercept is the original value. y

x 2

6

10

Exercise Set 1.6

23

10. A line through (6, −1) has the form y + 1 = m(x − 6). The intercepts are x = 6 + 1/m and y = −6m − 1. Set −(6 + 1/m)(6m + 1) = 3, or 36m2 + 15m + 1 = (12m + 1)(3m + 1) = 0 with roots m = −1/12, −1/3; thus y + 1 = −(1/3)(x − 6) and y + 1 = −(1/12)(x − 6). 11. (a) VI

(b) IV

(c) III

(d) V

(e) I

(f ) II

12. In all cases k must be positive, or negative values would appear in the chart. Only kx−3 decreases, so that must be f (x). Next, kx2 grows faster than kx3/2 , so that would be g(x), which grows faster than h(x) (to see this, consider ratios of successive values of the functions). Finally, experimentation (a spreadsheet is handy) for values of k yields (approximately) f (x) = 10x−3 , g(x) = x2 /2, h(x) = 2x1.5 . y

y

13. (a)

30

-2

x 1

-10

2

10 x -2

1

-40

2

-10

-30 y

y

(b)

10 4 2 6

x 2

-4

4

-2 x

-4 -4 y

(c)

2

4

y

2 1 1 x

x -1

2

3

1

2

3

-2

14. (a)

y

y 80

40 40 x -2

2

-40

x -2

2

24

Chapter 1

y

y

(b)

x

4 -2

2 -2

x -2

-4

2

-4

y

y

(c)

x

-2

2

4

4 -2

x -3

-1

-2

y

15. (a)

(b)

y

10

6

5

4 x

-2

2 x -5

-2

1 -2

-10

y

(c)

10 5 x -2

2 -5 -10

y

16. (a)

y

(b)

y

(c)

3 2

2 x

x 2

2 x 2

2

Exercise Set 1.6

25

y

17. (a)

y

(b)

80

6 20

x

-1 -20

2

1

2

3

4

5

x -3

1

-1

-80

y

(c) -3

-10

y

(d) x

40

1

20 x 1

2

3

4

5

-20 -40

-40

y

18. (a)

y

(b) 2

1

1

x

x

1

-6

-4

-2

2 -1 -2

y

(c)

y

(d)

4 x -2

2

4

-4

x -6

y

19. (a)

(b)

1.5

1

-2

y

0.5 0.5 -3

-2

x

x 1

1

3

-0.5 -1

-1

y

(c)

10 -1

30 10

x 1

-20

y

(d)

30

3

-2

x 1

-20

2

26

Chapter 1

y

20. (a)

y

(b)

10

2

x 1

-10

3

x

-4 -2

4

2

4

-4

y

(c)

y

(d)

106

10 x -1

1

5 x

-106

-2

21. y = x2 + 2x = (x + 1)2 − 1

2

y

10 5 x -2

2

y

22. (a)

y

(b) 1

1

x

x -2

-2

2

23. (a) N·m (c)

(b) k = 20 N·m

V (L)

0.25 2

P (N/m ) 80 × 10 (d)

2

1.0

0.5 3

40 × 10

3

20 × 10

1.5 3

13.3 × 10

2.0 3

10 × 103

P(N/m2) 30 20 10 V(m3) 10

20

24. If the side of the square base is x and the height of the container is y then V = x2 y = 100; minimize A = 2x2 + 4xy = 2x2 + 400/x. A graphing utility with a zoom feature suggests that the solution 1 is a cube of side 100 3 cm.

Exercise Set 1.6

27

25. (a) F = k/x2 so 0.0005 = k/(0.3)2 and k = 0.000045 N·m2 . (b) F = 0.000005 N F (c) 10-5

d 5

10

(d) When they approach one another, the force becomes inﬁnite; when they get far apart it tends to zero. 26. (a) 2000 = C/(4000)2 , so C = 3.2 × 1010 lb·mi2 (b) W = C/50002 = (3.2 × 1010 )/(25 × 106 ) = 1280 lb. W

(c)

(d)

No, but W is very small when x is large.

15000

5000 x 2000

8000

27. (a) II; y = 1, x = −1, 2 (c) IV; y = 2

(b) I; y = 0, x = −2, 3 (d) III; y = 0, x = −2

28. The denominator has roots x = ±1, so x2 − 1 is the denominator. To determine k use the point (0, −1) to get k = 1, y = 1/(x2 − 1). 29. Order the six trigonometric functions as sin, cos, tan, cot, sec, csc: (a) pos, pos, pos, pos, pos, pos (b) neg, zero, undef, zero, undef, neg (c) pos, neg, neg, neg, neg, pos (d) neg, pos, neg, neg, pos, neg (e) neg, neg, pos, pos, neg, neg (f ) neg, pos, neg, neg, pos, neg 30. (a) neg, zero, undef, zero, undef, neg (c) zero, neg, zero, undef, neg, undef (e) neg, neg, pos, pos, neg, neg

(b) pos, neg, neg, neg, neg, pos (d) pos, zero, undef, zero, undef, pos (f ) neg, neg, pos, pos, neg, neg

31. (a) sin(π − x) = sin x; 0.588 (c) sin(2π + x) = sin x; 0.588 (e) cos2 x = 1 − sin2 x; 0.655

(b) cos(−x) = cos x; 0.924 (d) cos(π − x) = − cos x; −0.924 (f ) sin2 2x = 4 sin2 x cos2 x = 4 sin2 x(1 − sin2 x); 0.905

32. (a) sin(3π + x) = − sin x; −0.588 (b) cos(−x − 2π) = cos x; 0.924 (c) sin(8π + x) = sin x; 0.588  (d) sin(x/2) = ± (1 − cos x)/2; use the negative sign for x small and negative; −0.195 (e) cos(3π + 3x) = −4 cos3 x + 3 cos x; −0.384 1 − cos2 x (f ) tan2 x = ; 0.172 cos2 x

28

Chapter 1

33. (a) −a (e) −b (i) 1/b

√ (d) ± 1 − a2

(c) −c

(b) b

√ (g) ±2b 1 − b2 (k) 1/c

(f ) −a (j) −1/a

(h) 2b2 − 1 (l) (1 − b)/2

34. (a) The distance is 36/360 = 1/10th of a great circle, so it is (1/10)2πr = 2, 513.27 mi. (b) 36/360 = 1/10 35. If the arc length is x, then solve the ratio

x 2πr = to get x ≈ 87, 458 km. 1 27.3

36. The distance travelled is equal to the length of that portion of the circumference of the wheel which touches the road, and that is the fraction 225/360 of a circumference, so a distance of (225/360)(2π)3 = 11.78 ft 37. The second quarter revolves twice (720◦ ) about its own center. 38. Add r to itself until you exceed 2πr; since 6r < 2πr < 7r, you can cut oﬀ 6 pieces of pie, but 3 2πr − 6r = 1 − times the original there’s not enough for a full seventh piece. Remaining pie is 2πr π pie. 39. (a) y = 3 sin(x/2)

(b) y = 4 cos 2x

(c) y = −5 sin 4x

40. (a) y = 1 + cos πx

(b) y = 1 + 2 sin x

(c) y = −5 cos 4x

41. (a) y = sin(x + π/2)

(b) y = 3 + 3 sin(2x/9)

(c) y = 1 + 2 sin(2(x − π/4))

(b) 2, 2, 0

(c) 1, 4π, 0

√ 42. V = 120 2 sin(120πt) 43. (a) 3, π/2, 0

y

y 3

y 3

2

2

1 c/2

x

x 2

4

1 x

-2 2c

-2

(c) 4, 6π, −6π

(b) 1/2, 2π/3, π/3

44. (a) 4, π, 0

y

y

y

4

0.4

2

2

x -2

3

9

f l -0.2

-4

4c 6c

8

c

x

x -2

i

-4

45. (a) A sin(ωt + θ) = A sin(ωt) cos θ + A cos(ωt) sin θ = A1 sin(ωt) + A2 cos(ωt)  (b) A1 = A cos θ, A2 = A sin θ, so A = A21 + A22 and θ = tan−1 (A2 /A1 ).

15c 21c 2 2

Exercise Set 1.7

29

√ 1 (c) A = 5 13/2, θ = tan−1 √ ; 2 3 √   5 13 −1 1 √ sin 2πt + tan x= 2 2 3

10

^

6

-10

46. three; x = 0, x = ±1.8955 3

-3

3

–3

EXERCISE SET 1.7 1. The sum of the squares for the residuals for line I is approximately 12 + 12 + 12 + 02 + 22 + 12 + 12 + 12 = 10, and the same for line II is approximately 02 + (0.4)2 + (1.2)2 + 02 + (2.2)2 + (0.6)2 + (0.2)2 + 02 = 6.84; line II is the regression line. 2. (a) (b) (c) (d)

The The The The

data data data data

appear appear appear appear

to to to to

be lie lie be

periodic, so a trigonometric model may be appropriate. near a parabola, so a quadratic model may be appropriate. near a line, so a linear model may be appropriate. randomly distributed, so no model seems to be appropriate.

3. Least squares line S = 1.5388t − 2842.9, correlation coeﬃcient 0.83409 S 240 230 220 210 200 190

t 1980 1983 1986 1989 1992 1995 1998

4. (a) p = 0.0154T + 4.19

(b) 3.42 atm

5. (a) Least squares line p = 0.0146T + 3.98, correlation coeﬃcient 0.9999 (b) p = 3.25 atm (c) T = −272◦ C 6. (a) p = 0.0203 T + 5.54, r = 0.9999 (b) T = −273 (c) 1.05 p = 0.0203(T )(1.1) + 5.54 and p = 0.0203T + 5.54, subtract to get .05p = 0.1(0.0203T ), p = 2(0.0203T ). But p = 0.0203T + 5.54, equate the right hand sides, get T = 5.54/0.0203 ≈ 273◦ C 7. (a) R = 0.00723T + 1.55

(b) T = −214◦ C

30

Chapter 1

8. (a) y = 0.0236x + 4.79

(b) T = −203◦ C

9. (a) S = 0.50179w − 0.00643

(b) S = 8, w = 16 lb

10. (a) S = 0.756w − 0.0133 (b) 2S = 0.756(w + 5) − 0.0133, subtract to get S = 5(0.756) = 3.78 11. (a) Let h denote the height in inches and y the number of rebounds per minute. Then y = 0.00630h − 0.266, r = 0.313 (b)

y

(c)

0.3

No, the data points are too widely scattered.

0.25 0.2 0.15 0.1 0.05 h 66 67 68 69 70 71 72 73 74 75

12. (a) Let h denote the height in inches and y the weight in pounds. Then y = 7.73h − 391 (b)

y

(c) 259 lb

280 260 240 220 200 180 160

h 72

76

80

84

88

13. (a) H ≈ 20000/110 ≈ 181 km/s/Mly (b) One light year is 9.408 × 1012 km and 1 d 1 9.408 × 1018 km t= = = = = 4.704 × 1017 s = 1.492 × 1010 years. v H 20km/s/Mly 20km/s (c) The Universe would be even older. 14. (a) f = 0.906m + 5.93

(b) f = 85.7

15. (a) P = 0.322t2 + 0.0671t + 0.00837

(b) P = 1.43 cm

16. The population is modeled by P = 0.0045833t2 − 16.378t + 14635; in the year 2000 the population would be P = 212, 200, 000. This is far short; in 1990 the population of the US was approximately 250,000,000.  

C 17. As in Example 4, a possible model is of the form T = D + A sin B t − . Since the longest B day has 993 minutes and the shortest has 706, take 2A = 993 − 706 = 287 or A = 143.5. The midpoint between the longest and shortest days is 849.5 minutes, so there is a vertical shift of D = 849.5. The period is about 365.25 days, so 2π/B = 365.25 or B = π/183. Note that the sine C function takes the value −1 when t − = −91.8125, and T is a minimum at about t = 0. Thus B π C π

is a model for the temperature. the phase shift ≈ 91.5. Hence T = 849.5 + 143.5 sin t− B 183 2

Exercise Set 1.8

31

18. As in Example 4, a possible model for the fraction f of illumination is of the form 

 C . Since the greatest fraction of illumination is 1 and the least is 0, f = D + A sin B t − B 2A = 1, A = 1/2. The midpoint of the fraction of illumination is 1/2, so there is a vertical shift of D = 1/2. The period is approximately 30 days, so 2π/B = 30 or B = π/15. The phase shift is  

π 49 approximately 49/2, so C/B = 49/2, and f = 1/2 + 1/2 sin t− 15 2 √ 19. t = 0.445 d 2.3

0

25 0

20. (a) t = 0.373 r1.5

(b) 238,000 km

(c) 1.89 days

EXERCISE SET 1.8 1. (a) x + 1 = t = y − 1, y = x + 2

(c)

0 1 t x −1 0 y 1 2

(c)

t 0 x 1 y 0

y t=5

6 t=4 t=3 t=2 t=1

4 2 t=0

2 1 3

4 3 5

5 4 6

x 2

4

2. (a) x2 + y 2 = 1 y t = 0.5 t = 0.75

1

t = 0.25

0.2500 0.50 0.7500 1 0.7071 0.00 −0.7071 −1 0.7071 1.00 0.7071 0

t=0 x

t=1 -1

1

3. t = (x + 4)/3; y = 2x + 10

4. t = x + 3; y = 3x + 2, −3 ≤ x ≤ 0

y

y

12 (0, 2) x

x -8

3 2 4

6

(-3, -7)

32

Chapter 1

5

7. cos t = (x − 3)/2, sin t = (y − 2)/4; (x − 3)2 /4 + (y − 2)2 /16 = 1

6. t = x2 ; y = 2x2 + 4, x≥0

5. cos t = x/2, sin t = y/5; x2 /4 + y 2 /25 = 1

y

y

y

8 4

x -5

6 x

5

1 x 7

-5 -2

9. cos 2t = 1 − 2 sin2 t; x = 1 − 2y 2 , −1 ≤ y ≤ 1

8. sec2 t − tan2 t = 1; x2 − y 2 = 1, x ≤ −1 and y ≥ 0

10. t = (x − 3)/4; y = (x − 3)2 − 9 y

y

y

x

1 x

x -1

-1

1 (3, -9) -1

11. x/2 + y/3 = 1, 0 ≤ x ≤ 2, 0 ≤ y ≤ 3

12. y = x − 1, x ≥ 1, y ≥ 0

y

y

3

1 x

x

2

1

13. x = 5 cos t, y = −5 sin t, 0 ≤ t ≤ 2π

14. x = cos t, y = sin t, π ≤ t ≤ 3π/2 1

5

-7.5

7.5

-1

1

–1

-5

16. x = 2 cos t, y = 3 sin t, 0 ≤ t ≤ 2π

15. x = 2, y = t

3

2

-2 0

-1

2

3

–3

Exercise Set 1.8

33

17. x = t2 , y = t, −1 ≤ t ≤ 1

18. x = 1 + 4 cos t, y = −3 + 4 sin t, 0 ≤ t ≤ 2π 2

1 -7 0

8

1

-8

-1

19. (a) (b) (c) (d)

IV, because x always increases whereas y oscillates. II, because (x/2)2 + (y/3)2 = 1, an ellipse. V, because x2 + y 2 = t2 increases in magnitude while x and y keep changing sign. VI; examine the cases t < −1 and t > −1 and you see the curve lies in the ﬁrst, second and fourth quadrants only. (e) III because y > 0. (f ) I; since x and y are bounded, the answer must be I or II; but as t runs, say, from 0 to π, x goes directly from 2 to −2, but y goes from 0 to 1 to 0 to −1 and back to 0, which describes I but not II.

20. (a) from left to right (b) counterclockwise (c) counterclockwise (d) As t travels from −∞ to −1, the curve goes from (near) the origin in the third quadrant and travels up and left. As t travels from −1 to +∞ the curve comes from way down in the second quadrant, hits the origin at t = 0, and then makes the loop clockwise and ﬁnally approaches the origin again as t → +∞. (e) from left to right (f ) Starting, say, at (1, 0), the curve goes up into the ﬁrst quadrant, loops back through the origin and into the third quadrant, and then continues the ﬁgure-eight. 21. (a)

(b)

14

-35

t 0 x 0 y 1

1 2 5.5 8 1.5 3

3 4 5 4.5 −8 −32.5 5.5 9 13.5

8 0

√ (c) x = 0 when t = 0, 2 3. (e) at t = 2 22. (a)

√ (d) for 0 < t < 2 2

(b) y is always ≥ 1 since cos t ≤ 1

5

-2

14 0

(c) greater than 5, since cos t ≥ −1

34

Chapter 1

23. (a)

o

(b)

3

0

20

-1

O

-5

24. (a)

1

-2.3

6

(b)

1.7

2.3

-10

10

-1.7

^

x − x0 y − y0 = x1 − x0 y1 − y0 (b) Set t = 0 to get (x0 , y0 ); t = 1 for (x1 , y1 ).

25. (a) Eliminate t to get

(c) x = 1 + t, y = −2 + 6t (d) x = 2 − t, y = 4 − 6t 26. (a) x = −3 − 2t, y = −4 + 5t, 0 ≤ t ≤ 1

(b) x = at, y = b(1 − t), 0 ≤ t ≤ 1

27. (a) |R−P |2 = (x−x0 )2 +(y−y0 )2 = t2 [(x1 −x0 )2 +(y1 −y0 )2 ] and |Q−P |2 = (x1 −x0 )2 +(y1 −y0 )2 , so r = |R − P | = |Q − P |t = qt. (b) t = 1/2

(c) t = 3/4

28. x = 2 + t, y = −1 + 2t (b) (9/4, −1/2)

(a) (5/2, 0)

(c) (11/4, 1/2)

29. The two branches corresponding to −1 ≤ t ≤ 0 and 0 ≤ t ≤ 1 coincide. 30. (a) Eliminate

t − t0 y − y0 y1 − y0 to obtain = . t1 − t0 x − x0 x1 − x0

(b) from (x0 , y0 ) to (x1 , y1 ) (c) x = 3 − 2(t − 1), y = −1 + 5(t − 1)

5

0

-2

5

Exercise Set 1.8

31. (a)

35

y−d x−b = a c

(b)

y 3 2 1 x 1

2

3

32. (a) If a = 0 the line segment is vertical; if c = 0 it is horizontal. (b) The curve degenerates to the point (b, d). 33.

y 2

1

x 0.5

34.

1

x = 1/2 − 4t, y = 1/2 x = −1/2, y = 1/2 − 4(t − 1/4) x = −1/2 + 4(t − 1/2), y = −1/2 x = 1/2, y = −1/2 + 4(t − 3/4)

for 0 ≤ t ≤ 1/4 for 1/4 ≤ t ≤ 1/2 for 1/2 ≤ t ≤ 3/4 for 3/4 ≤ t ≤ 1

35. (a) x = 4 cos t, y = 3 sin t (b) x = −1 + 4 cos t, y = 2 + 3 sin t (c)

3

5

-4

4 -5 -3

-1

36. (a) t = x/(v0 cos α), so y = x tan α − gx2 /(2v02 cos2 α). y

(b) 10000 6000 2000

3

x 40000 80000

√ √ 37. (a) From Exercise 36, x = 400 2t, y = 400 2t − 4.9t2 . (b) 16,326.53 m (c) 65,306.12 m

36

Chapter 1

38. (a)

(b)

15

-25

25

15

-25

25

–15

(c)

–15

15

-25

15

25

-25

–15 a = 3, b = 2

25

–15 a = 2, b = 3

15

-25

25

–15 a = 2, b = 7

39. Assume that a = 0 and b = 0; eliminate the parameter to get (x − h)2 /a2 + (y − k)2 /b2 = 1. If |a| = |b| the curve is a circle with center (h, k) and radius |a|; if |a| = |b| the curve is an ellipse with center (h, k) and major axis parallel to the x-axis when |a| > |b|, or major axis parallel to the y-axis when |a| < |b|. (a) ellipses with a ﬁxed center and varying axes of symmetry (b) (assume a = 0 and b = 0) ellipses with varying center and ﬁxed axes of symmetry (c) circles of radius 1 with centers on the line y = x − 1

40. Refer to the diagram to get bθ = aφ, θ = aφ/b but θ − α = φ + π/2 so α = θ − φ − π/2 = (a/b − 1)φ − π/2 x = (a − b) cos φ − b sin α   a−b = (a − b) cos φ + b cos φ, b y = (a − b) sin φ − b cos α   a−b = (a − b) sin φ − b sin φ. b

y

a−b

␪ ␾

b␪ a␾ x

Chapter 1 Supplementary Exercises

41. (a)

a

37

y

x a

-a

-a

(b) Use b = a/4 in the equations of Exercise 40 to get x = 34 a cos φ + 14 a cos 3φ, y = 34 a sin φ − 14 a sin 3φ; but trigonometric identities yield cos 3φ = 4 cos3 φ − 3 cos φ, sin 3φ = 3 sin φ − 4 sin3 φ, so x = a cos3 φ, y = a sin3 φ. (c) x2/3 + y 2/3 = a2/3 (cos2 φ + sin2 φ) = a2/3 y

42. (a)

y

y 5

3

1

1 x

-1

x

1

3

-1

a=3

-5

x 5

1 -3 -1

(b)

a=5

a = 1/2

(x − a)2 + y 2 = (2a cos2 t − a)2 + (2a cos t sin t)2 = 4a2 cos4 t − 4a2 cos2 t + a2 + 4a2 cos2 t sin2 t = 4a2 cos4 t − 4a2 cos2 t + a2 + 4a2 cos2 t(1 − cos2 t) = a2 , a circle about (a, 0) of radius a

CHAPTER 1 SUPPLEMENTARY EXERCISES 1. 1940-45; the greatest ﬁve-year slope 2. (a) (b) (c) (d) (e) (f ) 3.

f (−1) = 3.3, g(3) = 2 x = −3, 3 x < −2, x > 3 the domain is −5 ≤ x ≤ 5 and the range is −5 ≤ y ≤ 4 the domain is −4 ≤ x ≤ 4.1, the range is −3 ≤ y ≤ 5 f (x) = 0 at x = −3, 5; g(x) = 0 at x = −3, 2 4.

T

x

70

50 40

t

t 0

2

4

6

5

8

13

38

Chapter 1

5. If the side has length x and height h, then V = 8 = x2 h, so h = 8/x2 . Then the cost C = 5x2 + 2(4)(xh) = 5x2 + 64/x. 6. Assume that the paint is applied in a thin veneer of uniform thickness, so that the quantity of paint to be used is proportional to the area covered. If P is the amount of paint to be used, P = kπr2 . The constant k depends on physical factors, such as the thickness of the paint, absorption of the wood, etc. y

7.

5

x -5

5

-1

8. Suppose the radius of the uncoated ball is r and that of the coated ball is r + h. Then the plastic has volume equal to the diﬀerence of the volumes, i.e. V = 43 π(r + h)3 − 43 πr3 = 43 πh[3r2 + 3rh + h2 ] in3 . 9. (a) The base has sides (10 − 2x)/2 and 6 − 2x, and the height is x, so V = (6 − 2x)(5 − x)x ft3 . (b) From the picture we see that x < 5 and 2x < 6, so 0 < x < 3. (c) 3.57 ft ×3.79 ft ×1.21 ft 10. {x = 0} and ∅ (the empty set) 11. f (g(x)) = (3x + 2)2 + 1, g(f (x)) = 3(x2 + 1) + 2, so 9x2 + 12x + 5 = 3x2 + 5, 6x2 + 12x = 0, x = 0, −2 12. (a) (3 − x)/x (b) no; f (g(x)) can be deﬁned at x = 1, whereas g, and therefore f ◦ g, requires x = 1 13. 1/(2 − x2 ) 15.

x f (x) g(x)

14. g(x) = x2 + 2x −4 −3 −2 −1 0 −1 2 1 3

2

1

(f ◦ g)(x)

0 3

2

−2 −3

−3 −1 −4

4 −3 −2 −1 1 4 −4 −2 (g ◦ f )(x) −1 −3 16. (a)

1

0 1

y = |x − 1|, y = |(−x) − 1| = |x + 1|, y = 2|x + 1|, y = 2|x + 1| − 3, y = −2|x + 1| + 3

3

4

4

−4

4

−2

0

−4 2

2 0

3 3 y

(b)

3

x -3

17. (a) even × odd = odd (c) even + odd is neither

-1 -1

2

(b) a square is even (d) odd × odd = even

Chapter 1 Supplementary Exercises

39

Ď&#x20AC; 3Ď&#x20AC; Ď&#x20AC; 5Ď&#x20AC; 7Ď&#x20AC; 11Ď&#x20AC; ,â&#x2C6;&#x2019; ,â&#x2C6;&#x2019; 18. (a) y = cos x â&#x2C6;&#x2019; 2 sin x cos x = (1 â&#x2C6;&#x2019; 2 sin x) cos x, so x = Âą , Âą , , 2 2 6 6 6 6         Ď&#x20AC; â&#x2C6;&#x161; Ď&#x20AC;   5Ď&#x20AC; â&#x2C6;&#x161; 3Ď&#x20AC; 7Ď&#x20AC; â&#x2C6;&#x161; 11Ď&#x20AC; â&#x2C6;&#x161; , 3/2 , , â&#x2C6;&#x2019; 3/2 , â&#x2C6;&#x2019; , â&#x2C6;&#x2019; 3/2 , â&#x2C6;&#x2019; , 3/2 (b) Âą , 0 , Âą , 0 , 2 2 6 6 6 6 19. (a) If x denotes the distance from A to the base of the tower, and y the distance from B to the base, then x2 +d2 = y 2 . Moreover h = x tan Îą = y tan Î˛, so d2 = y 2 â&#x2C6;&#x2019;x2 = h2 (cot2 Î˛â&#x2C6;&#x2019;cot2 Îą), d2 d2 tan2 Îą tan2 Î˛ d2 = = , which yields the result. h2 = 2 2 2 2 cot Î˛ â&#x2C6;&#x2019; cot Îą 1/ tan Î˛ â&#x2C6;&#x2019; 1/ tan Îą tan2 Îą â&#x2C6;&#x2019; tan2 Î˛ (b) 295.72 ft. y

20. (a) 60

20 t 100

300

-20

3Ď&#x20AC; 2Ď&#x20AC; (t â&#x2C6;&#x2019; 101) = , or t = 374.75, which is the same date as t = 9.75, so during the 365 2 night of January 10th-11th

(b) when

(c) from t = 0 to t = 70.58 and from t = 313.92 to t = 365 (the same date as t = 0) , for a total of about 122 days 21. When x = 0 the value of the green curve is higher than that of the blue curve, therefore the blue curve is given by y = 1 + 2 sin x. The points A, B, C, D are the points of intersection of the two curves, i.e. where 1+2 sin x = 2 sin(x/2)+2 cos(x/2). Let sin(x/2) = p, cos(x/2) = q. Then 2 sin x = 4 sin(x/2) cos(x/2), so the equation which yields the points of intersection becomes 1 + 4pq = 2p + 2q, 4pq â&#x2C6;&#x2019; 2p â&#x2C6;&#x2019; 2q + 1 = 0, (2p â&#x2C6;&#x2019; 1)(2q â&#x2C6;&#x2019; 1) = 0; thus whenever either sin(x/2) =â&#x2C6;&#x161;1/2 or cos(x/2) = 1/2, i.e. whenâ&#x2C6;&#x161;x/2 = Ď&#x20AC;/6, 5Ď&#x20AC;/6, ÂąĎ&#x20AC;/3. Thus A â&#x2C6;&#x161; has coordinates (â&#x2C6;&#x2019;2Ď&#x20AC;/3, 1 â&#x2C6;&#x2019; 3), B has â&#x2C6;&#x161; coordinates (Ď&#x20AC;/3, 1 + 3), C has coordinates (2Ď&#x20AC;/3, 1 + 3), and D has coordinates (5Ď&#x20AC;/3, 1 â&#x2C6;&#x2019; 3). 22. Let y = A + B sin(at + b). Since the maximum and minimum values of y are 35 and 5, A + B = 35 and A â&#x2C6;&#x2019; B = 5, A = 20, B = 15. The period is 12 hours, so 12a = 2Ď&#x20AC; and a = Ď&#x20AC;/6. The maximum occurs at t = 2, so 1 = sin(2a + b) = sin(Ď&#x20AC;/3 + b), Ď&#x20AC;/3 + b = Ď&#x20AC;/2, b = Ď&#x20AC;/2 â&#x2C6;&#x2019; Ď&#x20AC;/3 = Ď&#x20AC;/6 and y = 20 + 15 sin(Ď&#x20AC;t/6 + Ď&#x20AC;/6). 23. (a) The circle of radius 1 centered at (a, a2 ); therefore, the family of all circles of radius 1 with centers on the parabola y = x2 . (b) All parabolas which open up, have latus rectum equal to 1 and vertex on the line y = x/2. 24. (a) x = f (1 â&#x2C6;&#x2019; t), y = g(1 â&#x2C6;&#x2019; t)

y

25. 1

x -2

1

-2

2

40

Chapter 1

26. Let y = ax2 + bx + c. Then 4a + 2b + c = 0, 64a + 8b + c = 18, 64a − 8b + c = 18, from which b = 0 3 2 x − 65 . and 60a = 18, or ﬁnally y = 10 y

27. 2 1

x -1

1

2

-1 -2

28. (a) R = R0 is the R-intercept, R0 k is the slope, and T = −1/k is the T -intercept (b) −1/k = −273, or k = 1/273 (c) 1.1 = R0 (1 + 20/273), or R0 = 1.025 (d) T = 126.55◦ C 29. d =

 √ (x − 1)2 + ( x − 2)2 ;

30. d =

d = 9.1 at x = 1.358094

 (x − 1)2 + 1/x2 ;

d = 0.82 at x = 1.380278 y

y 2

2

1.6 1

1.2 1 0.8 0.5 1

x 1

2

x 2

3

31. w = 63.9V , w = 63.9πh2 (5/2 − h/3); h = 0.48 ft when w = 108 lb 32. (a)

(b)

W

w = 63.9πh2 (5/2 − h/3); at h = 5/2, w = 2091.12 lb

3000

1000 h 1

33. (a)

3

5

34. (a)

N

T

200

100 10 t 10

30

50

(b) N = 80 when t = 9.35 yrs (c) 220 sheep

v 20

(b) T = 17◦ F, 27◦ F, 32◦ F

Chapter 1 Supplementary Exercises

35. (a)

41

(b) T = 3◦ F, −11◦ F, −18◦ F, −22◦ F

WCI 20

(c) v = 35, 19, 12, 7 mi/h v 20

40

-20

36. The domain is the set of all x, the range is −0.1746 ≤ y ≤ 0.1227. 37. The domain is the set −0.7245 ≤ x ≤ 1.2207, the range is −1.0551 ≤ y ≤ 1.4902. 38. (a) The potato is done in the interval 27.65 < t < 32.71. (b) 91.54 min. v

39. (a) 20

5 t 1

2

3

4

5

(b) As t → ∞, (0.273)t → 0, and thus v → 24.61 ft/s. (c) For large t the velocity approaches c. (d) No; but it comes very close (arbitrarily close). (e) 3.013 s 40. (a) y = −0.01716428571x + 1.433827619 41. (a)

1.90

1.92

1.94

1.96

1.98

2.00

2.02

2.04

2.06

2.08

2.10

3.4161

3.4639

3.5100

3.5543

3.5967

3.6372

3.6756

3.7119

3.7459

3.7775

3.8068

(b) y = 1.9589x − 0.2910 (c) y − 3.6372 = 1.9589(x − 2), or y = 1.9589x − 0.2806 (d) As one zooms in on the point (2, f (2)) the two curves seem to converge to one line.

3.8

1.9

2.2 3.4

42. (a)

−0.10 0.9950

−0.08 0.9968

(b) y = − 12 x2 + 1

−0.06 0.9982

−0.04 0.9992

−0.02 0.9998

0.00 1.0000

0.02 0.9998

0.04 0.9992

0.06 0.9982

0.08 0.9968

0.10 0.9950

42

Chapter 1

(c) y = − 12 x2 + 1 (d) As one zooms in on the point (0, f (0)) the two curves seem to converge to one curve.

1.2

-1.7

1.7

-0.8

43. The data are periodic, so it is reasonable that a trigonometric function might approximate them. 

 C . Since the highest level is 1.032 A possible model is of the form T = D + A sin B t − B meters and the lowest is 0.045, take 2A = 1.032 − 0.042 = 0.990 or A = 0.495. The midpoint between the lowest and highest levels is 0.537 meters, so there is a vertical shift of D = 0.537. C The period is about 12 hours, so 2π/B = 12 or B = π/6. The phase shift ≈ 6.5. Hence B

π (t − 6.5) is a model for the temperature. T = 0.537 + 0.495 sin 6 T 1. 0.8 0.6 0.4 0.2 t 10 20 30 40 50 60 70

CHAPTER 1 HORIZON MODULE 1. (a) 0.25, 6.25 × 10−2 , 3.91 × 10−3 , 1.53 × 10−5 , 2.32 × 10−10 , 5.42 × 10−20 , 2.94 × 10−39 , 8.64 × 10−78 , 7.46 × 10−155 , 5.56 × 10−309 ; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1; 4, 16, 256, 65536, 4.29 × 109 , 1.84 × 1019 , 3.40 × 1038 , 1.16 × 1077 , 1.34 × 10154 , 1.80 × 10308 2. 1, 3, 2.3333333, 2.23809524, 2.23606890, 2.23606798, . . . 3. (a) 4. (a) (b) (c) (d)

1 1 1 1 1 1 , , , , , 2 4 8 16 32 64

(b) yn =

1 2n

yn+1 = 1.05yn y0 =\$1000, y1 =\$1050, y2 =\$1102.50, y3 =\$1157.62, y4 =\$1215.51, y5 =\$1276.28 yn+1 = 1.05yn for n ≥ 1 yn = (1.05)n 1000; y15 =\$2078.93

Chapter 1 Horizon Module

43

5. (a) x1/2 , x1/4 , x1/8 , x1/16 , x1/32 (b) They tend to the horizontal line y = 1, with a hole at x = 0. 1.8

0

3 0

6. (a) (b)

(c)

1 2

2 3

3 5

5 8

8 13

13 21

21 34

34 55

55 89

89 144

1, 2, 3, 5, 8, 13, 21, 34, 55, 89; each new numerator is the sum of the previous two numerators. 144 233 377 610 987 1597 2584 4181 6765 10946 , , , , , , , , , 233 377 610 987 1597 2584 4181 6765 10946 17711

(d) F0 = 1, F1 = 1, Fn = Fn−1 + Fn−2 for n ≥ 2. (e) the positive solution 7. (a) y1 = cr, y2 = cy1 = cr2 , y3 = cr3 , y4 = cr4

(b) yn = crn

(c) If r = 1 then yn = c for all n; if r < 1 then yn tends to zero; if r > 1, then yn gets ever larger (tends to +∞). 8. The ﬁrst point on the curve is (c, kc(1 − c)), so y1 = kc(1 − c) and hence y1 is the ﬁrst iterate. The point on the line to the right of this point has equal coordinates (y1 , y1 ), and so the point above it on the curve has coordinates (y1 , ky1 (1 − y1 )); thus y2 = ky1 (1 − y1 ), and y2 is the second iterate, etc. 9. (a) 0.261, 0.559, 0.715, 0.591, 0.701 (b) It appears to approach a point somewhere near 0.65.

sol1
sol1